There is a number of recurrence relations between the Whittaker functions,
for example [1]}, [2]}
\(\begin{array}{c}2\mu \left[ \mathrm {M}_{\kappa -1/2,\mu -1/2}\left( t\right) -\mathrm {M}_{\kappa +1/2,\mu -1/2}\left( t\right) \right] =t^{1/2}\mathrm {M}_{\kappa ,\mu }\left( t\right) , \\\left( \kappa +\mu \right) \mathrm {W}_{\kappa -1/2,\mu }\left( t\right) +\mathrm {W}_{\kappa +1/2,\mu }\left( t\right) =t^{1/2}\mathrm {W}_{\kappa ,\mu +1/2}\left( t\right) ,\end{array}\) 
